非线性薛定谔方程的非线性镜像方法：边界对单孤子的吸收/发射
Nonlinear mirror image method for nonlinear Schrödinger equation: Absorption/emission of one soliton by a boundary
AI总结：
本文通过非线性镜像方法和Bäcklund变换，研究半直线上时间依赖边界条件的聚焦非线性薛定谔方程，发现孤子可被边界吸收或发射，这是时间依赖可积边界条件的独特特征。
摘要：
我们利用Bäcklund变换，沿着非线性镜像方法的思路，分析了半直线上具有时间依赖边界条件的聚焦非线性薛定谔方程。在$x=0$处具有这种时间依赖边界条件所带来的困难，通过改变方法的视角，将Bäcklund变换固定在无穷远处，并将其在$x=0$处的值与时间依赖的反射矩阵相关联来克服。我们详细回顾了可积边界条件各个方面之间的相互作用，以描绘该领域的图景。我们找到了两类可能的解。一类与Robin边界条件的情况非常相似，其中孤子在边界处被反射，这是与另一半直线上的镜像有效相互作用的结果。新的解区域支持一个孤子的存在，该孤子不在边界处反射，而是可以被边界吸收或发射。我们证明这是时间依赖可积边界条件的一个独特特征。
英文摘要：
We perform the analysis of the focusing nonlinear Schrödinger equation on the half-line with time-dependent boundary conditions along the lines of the nonlinear method of images with the help of Bäcklund transformations. The difficulty arising from having such time-dependent boundary conditions at $x=0$ is overcome by changing the viewpoint of the method and fixing the Bäcklund transformation at infinity as well as relating its value at $x=0$ to a time-dependent reflection matrix. The interplay between the various aspects of integrable boundary conditions is reviewed in detail to brush a picture of the area. We find two possible classes of solutions. One is very similar to the case of Robin boundary conditions whereby solitons are reflected at the boundary, as a result of an effective interaction with their images on the other half-line. The new regime of solutions supports the existence of one soliton that is not reflected at the boundary but can be either absorbed or emitted by it. We demonstrate that this is a unique feature of time-dependent integrable boundary conditions.
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